Automated Bayesian Inference for PDE-constrained Inverse Problems Ivan Yashchuk VTT Technical Research Centre of Finland, Materials Modeling group Aalto University, Probabilistic Machine Learning group July 9, 2019 1
Model + Simulation + Data 2
Bayesian inference summary A generative physical model: ❳ ∼ F θ , being F θ a PDE system F ( ❳ ; θ ) = 0 Having observed some data ② define a likelihood p ( ② | ❳ , θ ) and combine with a prior distribution p ( θ ) to obtain the posterior p ( ② | ❳ , θ ) p ( θ ) p ( θ | ② ) = � p ( ② | ❳ , θ ) p ( θ ) d θ 3
Approximate methods: MCMC (lots of variants), Variational method (lots of variants), Laplace approximation, neural nets, sparse grid, etc. 4
Bayesian inference challenges and opportunities Because of ... high-dimensionality of the target we need to utilize derivative information repeated PDE solution we need to utilize the resources efficiently, use advanced solvers not everyone can do this we need to lower the barriers of skills needed by providing the tools 5
My PhD Components • Automatic differentiation for PDE systems Automated generation of adjoint problems and efficient integration of the adjoint solution into AD framework • Bayesian analysis for inverse problems Assessing prediction reliability and accuracy guarantees • Case studies Focus on Additive Manufacturing Large-scale multiphysics simulations • Software development 6
Additive Manufacturing 7
Additive Manufacturing 8
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